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In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. The resolvent captures the spectral properties of an operator in the analytic structure of the resolvent. Given an operator , the resolvent may be defined as : Among other uses, the resolvent may be used to solve the inhomogeneous Fredholm integral equations; a commonly used approach is a series solution, the Liouville-Neumann series. The resolvent of can be used to directly obtain information about the spectral decomposition of . For example, suppose is an isolated eigenvalue in the spectrum of . That is, suppose there exists a simple closed curve in the complex plane that separates from the rest of the spectrum of . Then the residue : defines a projection operator onto the eigenspace of . The Hille-Yosida theorem relates the resolvent through a Laplace transform to an integral over the one-parameter group of transformations generated by . Thus, for example, if is Hermitian, then is a one-parameter group of unitary operators. The resolvent can be expressed as their Laplace transform integral : ==History== The first major use of the resolvent operator was by Ivar Fredholm, in a landmark 1903 paper in ''Acta Mathematica'' that helped establish modern operator theory. The name ''resolvent'' was given by David Hilbert. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Resolvent formalism」の詳細全文を読む スポンサード リンク
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